ALGEBRA-3

If  x+5y = 1            find  ^x/y             

    x-3y     4

solution

x+5y = 1            

x-3y    4

1(x-3y) = 4(x+5y)           [after cross multiplying]

x-3y = 4x+20y

x - 4x = 20y + 3y

-3x = 23y

¹-3x = 23y

₁-3      -3

 X = -23y/3

x/y = -23/3        [after dividing by y on both sides]

x = -23/3

y    

Hence ^x/_y = -23/3  answer.           

TRY THIS......................

If  x+2y = 1            find  ^x/y             

   x- 2y     5

LOGARITHMS-3

Find x if Log_x0.00001 = -5

solution

Log_x0.00001 = -5

0.00001  = x^-5 .  [0.000001  = 10^-5  ]

10^-5 = x^-5 .

10⁵ = x⁵ .  { equal powers cancel out}

x = 10

Hence x = 10

TRY THIS..................

Find x if Log_x256 = 8

FUNCTIONS-1

If f(x) = |x - 24| evaluate f(-27)

solution

f(x) = |x - 24|

f(-27) = |-27 - 24|

f(-27) = |-51| = 51 since any number out of absolute signs is +ve.

Hence f(-27) = 51

TRY THIS..................

If f(x) = |x - 40| evaluate f(-15)

RATIOS-1

Given the ratios M:N = 16: 8 and N:P = 4:7 Find the ratio M:P.

solution

M = M x N

P      N     P

M =   16 x 4¹

P      ₂ 8     7

M =      16

P        (7x2)

M =       ⁸16  =  8

P           ₇14      7

Hence M:P=8:7

TRY THIS...............................

Given the ratios M:N = 20: 8 and N:P = 12:7 Find the ratio M:P.

ABSOLUTE VALUE EQUATIONS-1

If │2x – 13 │= 21; Find  x

Solution

±(2x – 13 )= 21

2x -13= 21    OR   –(2x-13) = 21

2x – 13 = 21    OR   -2x+13= 21

2x = 21 + 13    OR   -2x= 21-13

2x = 34     OR   -2x  =  8

2      2                2        2

x = 17  OR  x = 4

Hence x = 17 OR x = 4     

TRY THIS...............................

If │4x – 16 │= 32; Find  x

PERCENTAGE PROFIT-1

A man got a profit of 200/= after selling an item for 8000/=. Find the percentage profit.

Solution

%’ge profit = Profit   X  100    [where B. P. represents Buying Price.]

                      B.P.

%’ge profit = 200   X  100   

                    8000

%’ge profit = 2.5%   

                       

Hence Percentage Profit is 2.5%

TRY THIS……………………………

A man got a profit of 10,000/= after selling an item for 600,000/=. Find the percentage profit.

DIFFERENCE OF 2 SQUARES-2

Factorize 1-V²

Solution

We use difference of two squares a² – b² = (a - b)(a + b)

1-V² = 1²-V²       

        = (1 - V)(1 + V)

Hence 1 - V² = (1 - V)(1 + V)

TRY THIS…………….

Factorize 25 - W²

LOGARITHMS-2

If log 3= 0.4771; find the value of log 90,000 without using tables.

solution

log90,000=log(10,000 x 9)

=log10,000 + log9

=log10⁴ – log3²

=4log10 – 2log3

=(4x1) –  2(0.4771)

=4– 0.9542

= 3.0458

Hence log 90,000 = 3.0458

TRY THIS……………

If log 3= 0.4771; find the value of log 8,100 without using tables.

EXPONENTIALS-1

If a² = 3; find the value of a⁶.

solution

a⁶ = (a²)³.

a⁶ = (3)³.   [ since a² = 3]

a⁶ = 3 x 3 x 3

a⁶ = 27  answer

TRY THIS.........................

If a² = 13; find the value of a⁴.

APPROXIMATION-2

Estimate the value of 4143  ÷ 52

Solution

= 4123     

     52

= 4000      [ since 4143 ≈ 4000   and   52 ≈ 50 ]

     50

= 80

TRY THIS...................................

Estimate the value of 7203  ÷ 19